Real-rooted atlas

A126351 → reverse → θ+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-2 - 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(-x - x^{2}\right)\,P_{n-2}'(x) + \left(3 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2410317
383827477
416130165565372
53242285550010061915
6641330405336401225162710481
712841181820723604119002604245860814
825612610790051422961009053250849983529372939
951238342334935816820783300348642779108880486102409837
10102411605013972534530680572082533804541030029168960148506501116360786
11204835019857630872451640439988300305281681218409526978163385802365084712116393205
12409610546902357764513026613627057530526168313613308895638205024642084363635036179108013865549453
138192317226295892615682767540178628450021564723181370008640495946880107189082141341201134133535081352146713065156
141638495331703883288533541694520115711664251723884762613475953491602924608016298310602745743002915712819328407689516661554190360453
Technical details
Node id
n-d72031dd4f362d3c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9be755efdd908b94

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2,     2;
  2:   4,    10,      3;
  3:   8,    38,     27,      4;
  4:  16,   130,    165,     56,      5;
  5:  32,   422,    855,    500,    100,      6;
  6:  64,  1330,   4053,   3640,   1225,    162,     7;
  7: 128,  4118,  18207,  23604,  11900,   2604,   245,    8;
  8: 256, 12610,  79005, 142296, 100905,  32508,  4998,  352,   9;
  9: 512, 38342, 334935, 816820, 783300, 348642, 77910, 8880, 486, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(2+2*x+x^2)*P(n-2,x) - (x+x^2)*d/dx P(n-2,x) + (3+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(2 + 2*x + x^2)*P[n - 2, x] - (x + x^2)*D[P[n - 2, x], x] + (3 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(2 + 2*x + x^2)*v[i - 2] - (x + x^2)*deriv(v[i - 2], x) + (3 + 2*x)*v[i - 1] + x*deriv(v[i - 1], x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  7. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.